For a polynomial p(z) of degree n, having all zeros in |z|< k, k< 1, Dewan et al [K. K. Dewan, N. Singh and A. Mir, Extension of some polynomial inequalities to the polar derivative, J. Math. Anal. Appl. 352 (2009) 807-815] obtained inequality between the polar derivative of p(z) and maximum modulus of p(z). In this paper we improve and extend the above inequality. Our result generalizes certai...

Using the Ratios Conjecture, we write down precise formulas with lower order terms for one and two level densities of zeros quadratic Dirichlet L–functions over function fields. We denote various arising as Type-0, Type-I Type-II contributions. When support Fourier transform test is sufficiently restricted, rigorously compute Type-0 confirm that they match conjectured answer. restrictions on ar...

Let \(\Re_n\) be the set of all rational functions type \(r(z) = p(z)/w(z),\) where \(p(z)\) is a polynomial degree at most \(n\) and \(w(z) \prod_{j=1}^{n}(z-a_j)\), \(|a_j|&gt;1\) for \(1\leq j\leq n\). In this paper, we up some results with fixed poles restricted zeros. The obtained bring forth generalizations refinements known inequalities in turn produce as well.

Many optimization algorithms involve repeated processing of a fixed set of linear constraints. If we pre-process the constraint matrix A to make it sparser, algebraic operations should become faster. In many applications there is a priori information about the likelihood that each column will appear in a basis, which can be expressed as weights on the columns. This leads to considering the Weig...

We consider the number of zeros of the integral $I(h) = oint_{Gamma_h} omega$ of real polynomial form $omega$ of degree not greater than $n$ over a family of vanishing cycles on curves $Gamma_h:$ $y^2+3x^2-x^6=h$, where the integral is considered as a function of the parameter $h$. We prove that the number of zeros of $I(h)$, for $0 < h < 2$, is bounded above by $2[frac{n-1}{2}]+1$.

Integer compositions with certain colored parts were introduced by Andrews in 2007 to address a number-theoretic problem. allowing zero as some Ouvry and Polychronakos 2019. We give bijection between these two varieties of determine various combinatorial properties multicompositions. In particular, we the count multicompositions number all parts, positive zeros. Then, working from three types r...

In this paper we obtain new results concerning maximum modulus of the polar derivative of a polynomial with restricted zeros. Our results generalize and refine upon the results of Aziz and Shah [An integral mean estimate for polynomial, Indian J. Pure Appl. Math. 28 (1997) 1413–1419] and Gardner, Govil and Weems [Some result concerning rate of growth of polynomials, East J. Apporox. 10(2004) 30...

The connection between random matrix theory and the Riemann zeta function was established in 1973 when Montgomery, who had conjectured the 2-point correlations of the Riemann zeros, and Dyson, who was interested in similar statistics of the eigenvalues of ensembles of unitary matrices, realized that the formulae they had discovered independently were in fact identical in a natural asymptotic li...

‎let $p(z)=z^s h(z)$ where $h(z)$ is a polynomial‎ ‎of degree at most $n-s$ having all its zeros in $|z|geq k$ or in $|z|leq k$‎. ‎in this paper we obtain some new results about the dependence of $|p(rz)|$ on $|p(rz)| $ for $r^2leq rrleq k^2$‎, ‎$k^2 leq rrleq r^2$ and for $rleq r leq k$‎. ‎our results refine and generalize certain well-known polynomial inequalities‎.